groups and physics coadjoint action momentum
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...Note that the system proposed to the child is not without flaws. It works properly only if the objects provided are those that came with the game itself. It should be noted that the "cylinders" compartment allows inserting cylinders of the same radius but different lengths, as well as paperclips, a bottle nipple, a cabinet key, etc...
...Logically, the child undergoing this group learning can deduce that a cabinet key and a cylinder are objects of the same kind. This is true, in the sense that these objects share the common property "which fits into that hole".
...My daughter, when she was much younger, conducted very interesting experiments on groups using the input of my computer's drive as an entry window, as a sieve. It required a complete disassembly, after being disabled, to reconstruct her reasoning, which was ultimately logical. Today, she remains very interested in groups. But these are not the same ones, and these choices do not please me much.
...Let's return to our child from earlier. A few months later, you will introduce the same child to the Galilean group, by throwing objects at him and encouraging him to catch them. Then, the shape of the objects is no longer important, what matters are their movements. We change groups. For the baby, the object (of sufficiently small size to be able to catch it) becomes assimilated to its center of gravity. It is a "point-mass", a "material point". The Galilean group manages the dynamics of material points .
The classification is then done according to the types of movements.
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This, I can catch.
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This, I cannot catch.
...Appreciating the initial conditions, position and velocity vector, at a glance, the baby must recognize the type of movement he is facing, and anticipate.
...As he grows up, he may play tennis, always using the Galilean group, which, like the Euclidean group, is a square matrix. As tennis balls move at speeds obviously much lower than the speed of light, he will not need to resort to the Poincaré group (another square matrix, which manages the relativistic movements of material points) .
...This being said, the material points managed by the Galilean group are no longer the points of the Euclidean group, they have attributes. The specialist in mathematical physics who manipulates the Galilean group no longer speaks of material points, but of movements. What he seeks to classify are movements. Then we come up against an essential aspect of elementary particle physics: we seek to associate a phenomenological description:
Tell me how your movements are, and I will tell you what you are.
...We no longer seek to know "what a particle is made of", but how it behaves. Thus, neutral particles do not behave in the same way as electrically charged particles. They belong to different species. They have different attributes.
...Our baby from earlier, now a high-energy physicist, may examine daily photographs taken in a bubble chamber.
Photographs obtained using a bubble chamber (Schem..) ...Left photo: a proton, a neutron and an electron have passed through the field of the chamber. The chamber is subjected to a magnetic field perpendicular to the plane of the figure. The neutron, without electric charge, is unaffected. It goes straight ahead. The charged particles have very different radii of rotation (Larmor radius). The light electrons turn much more sharply in a magnetic field perpendicular to the direction of their trajectory than the heavy protons. And both turn in opposite directions.
Set of movements, in a magnetic field, managed by....
But this group does not exist yet. If you found it, you would make many happy.
...Still, our man, examining his photos, detects trajectories belonging to different sets of movements.
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This one goes straight, it's a neutron.
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This one turns to the right slowly, it's a proton.
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This one turns sharply to the left, it's an electron.
...Right photo: an electron and an anti-electron, born from the same radiation (dashed line), adopt symmetrical trajectories, indicating that they have opposite electric charges.
Behavioral classification of species, as sets of movements.
The momentum.
...This purely geometric object can be considered as a set of attributes. Let us set aside charged particles, we will come back to them later. A "relativistic material point" has attributes grouped according to what the mathematician Jean-Marie Souriau, leader in mathematical physics, calls an object called moment linked to the Poincaré group.
The attributes of the relativistic material point are called:
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Energy E
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Momentum p - Spinning l (related to spin)
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Passage f
The "momentum" is thus:
**J **= { E , p , **l **, **f **}
**Note in passing **:
...Therefore, we have become accustomed, in everything that follows, to denote scalar quantities with thin letters and non-scalar (square matrices, row matrices, column matrices) with bold letters.
...Precision: we can then perform all row-column matrix multiplications by manipulating these thin or bold letters, which is extremely convenient. Let's give an example. The action of the element of the 2d Euclidean group was written:
By introducing:
we will obtain the more compact writing:
g then appears as a matrix, itself composed of submatrices:
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a is a square matrix of format (2,2).
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c is a column vector (the translation vector) of format (2,1).
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0 is a row vector of format (1,2) :
In general, the bold 0 are either row vectors or column vectors.
The action is then written:
Given that **a r **means a x r (but we eventually neglect the sign of the matrix multiplication).
End of the note, back to the theme of momentum. Let's return to the expression of it in the case of the relativistic material point.
**J **= { E , p , **l **, **f **}
E is a scalar (the energy).
**p **is the momentum vector
l and **f **(bold letters) are other vectors (lx,ly,lz) and (fx,fy,fz): the "spinning" and the "passage".
...Through our personal work that we will present in this sub-site Geometrical Physics B (the dynamic groups of physics), the problem will be precisely to make other "attributes" of elementary particles emerge, as a component of a richer moment (the charges: electric, baryonic, lepton, tauonic and the gyromagnetic coefficient).
...It was Souriau who, in the 1970s, built the method allowing the components of the momentum of a material point to appear, starting from the group that manages it (in the relativistic case, it is the Poincaré group). See the book: Structure of Dynamical Systems. Dunod 1973
Strongly recommended aspirin.
...It is then difficult to go further without implementing a fairly extensive mathematical equipment, otherwise complicated. Perhaps we will do this later, on the site, if there are enthusiasts, in the style "Everything you always wanted to know about groups without ever daring to ask".
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